nLab Tate K-theory




Tate K-theory is the elliptic cohomology theory associated with the Tate curve (the Tate elliptic curve over the Laurent series ring ((q))\mathbb{Z}((q)) ) (AHS 01, 2.7, Lurie 09, 4.3)

The corresponding elliptic genus is the Witten genus (AHS 01, Sec. 2.7).

Plain version

The underlying cohomology theory is given by Laurent series in topological K-theory, and equivalently by completion of circle group-equivariant K-theory of free loop spaces ()\mathcal{L}(-) (KM 04, Section 5, see also Lurie 09, Section 5.2):

Ell Tate(X) K(X)((q)) K S 1(X) K S 1(*)((q)). \begin{aligned} Ell_{Tate}(X) & \simeq \; K(X)((q)) \\ & \simeq \; K_{S^1}(\mathcal{L}X) \otimes_{K_{S^1}(\ast)} \mathbb{Z}((q)) \,. \end{aligned}

(For more exposition see also Dove 19, Sec. 6.2).

Equivariant version

The equivariant version of Tate K-theory is a form of equivariant elliptic cohomology. For GG a finite group and XX a topological G-space it comes down (Ganter 07, Def. 3.1, Ganter 13, Def. 2.6, review in Dove 19, Def. 6.16) to the sub-ring

Ell Tate((XG))[g]K C g(X g)((q 1/|g|)) Ell_{Tate} \big( \prec (X \sslash G) \big) \;\subset\; \underset{[g]}{\bigoplus} K_{C_g}(X^g)(( q^{1/\left\vert g\right\vert} ))

of the direct sum, over conjugacy classes of group elements gg, of Laurent polynomials with coefficients in equivariant K-theory-groups on the gg-fixed loci for equivariance group the centralizer of gg

on those elements which satisfy the rotation condition:

Rotation condition. The C gC_g-equivariant vector bundles V jV_j which form the coefficient of q j/|g|q^{j/\left\vert g \right\vert } are such that gg acts on them by multiplication with exp(2πij|g|)\exp\big( 2 \pi i \frac{j}{\left\vert g \right\vert} \big).

This rotation condition may be understood more intrinsically (this is made explicit on p. 63 of Dove 19) as that in implied by the orbifold K-theory on Huan's inertia orbifold (Huan 18), induced by the nature of the quotient groups

Λ gC g×(g 1,1) \Lambda_g \;\coloneqq\; \frac{C_g \times \mathbb{R}}{ \langle (g^{-1},1) \rangle }

appearing there. This quotient exactly equates the action of gGg \in G with that of a rotation of S 1S^1.

Hence, in generalization of twisted ad-equivariant K-theory there is twisted ad-equivariant Tate K-theory (an equivariant elliptic cohomology theory) relating to the Verlinde ring of positive energy loop group representations (Lurie 09, Sec. 5.2, Luecke 19, Cor. 3.2.5, Dove 19).


Relation to elliptic genus

The Ochanine genus lifts to a homomorphism of ring spectra MSpinKO((q))M Spin \to KO((q)) from spin structure cobordism cohomology theory to Tate K-theory (Kreck-Stolz 93, lemma 5.8, lemma 5.4). This is the spin-orientation of elliptic cohomology


As elliptic cohomology over the Tate sphere:

As completed S 1S^1-equivariant K-theory of free loop space

As a form of equivariant elliptic cohomology (twisted ad-equivariant Tate K-theory):

with a streamlined account in

and some general abstract clarification on Huan's inertia orbifold in:


  • Zhen Huan, Quasi-elliptic cohomology theory and the twisted, twisted Real theories, talk at Perimeter Institute (2020) [video: doi:10.48660/20050059]

For simply-connected compact Lie groups:

For finite groups:

Generalization to twisted equivariant K-theory

and further to KR-theory:

Last revised on November 3, 2023 at 06:47:17. See the history of this page for a list of all contributions to it.